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calculus_jy
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in linear momentum, considering the collision of 2 particles to deudce that the momentum in a closed system is concerved
(W_1 + F_12)t = m.v_f1 - m.v_01 ...(i)
where
W_1=external force during impact
F_12=force acting on body 1 by body 2 in collision
t=impacting time
m.v_f1= momentum of body 1 after collision
m.v_01=momentum of body 1 before collision
similiarly
(W_2 + F_21)t = m.v_f2 - m.v_02 ...(ii)
adding 1 and 2
since F_12 = -F_21 by Newton's third law
therefore (W_1+W_2)t = (m.v_f1 +m.v_f2) - (m.v_01 + m.v_02)
if the particles is isolated system, W_1+W_2=0 and the linear momentum of the system is conserved
similary result can be deueced by adding more of these equations of same form of (i) and (ii)
with calculus
F=dp/dt=d(mv)/dt= m. dv/dt= ma
with no external force a=o and linear momentum is conseved
I get how this works similiarly with rotational dynamics only in COLLISION
in the first method, conservation of angular momentum closed system durning collsion can be dudeuced by the 1st method by replacing
W_1 by torque by exteral force
F_12 by T_12= torque created on body(or disc) 1 by body 2
m.v_f1 by I.w_f1= angular momentum of body after collision
m.v_01 by I.w_01= angular momentum of body before collision
and by calculus
T= dL/dt = d(Iw)/dt =I. dw/dt = Ia
this is what i don't get: in the case of a ice skater, pulling arms in during rotation, I(moment of inertia) is changed, so how is conservation of momentum explained with calculus method when I of the system changes and i don't get at all how the non calculus method explains the conservation in this case (i only get it if it is during impact in rotational but in linear if totally understand how this method explains the conservation whether it impact of no impact)
(W_1 + F_12)t = m.v_f1 - m.v_01 ...(i)
where
W_1=external force during impact
F_12=force acting on body 1 by body 2 in collision
t=impacting time
m.v_f1= momentum of body 1 after collision
m.v_01=momentum of body 1 before collision
similiarly
(W_2 + F_21)t = m.v_f2 - m.v_02 ...(ii)
adding 1 and 2
since F_12 = -F_21 by Newton's third law
therefore (W_1+W_2)t = (m.v_f1 +m.v_f2) - (m.v_01 + m.v_02)
if the particles is isolated system, W_1+W_2=0 and the linear momentum of the system is conserved
similary result can be deueced by adding more of these equations of same form of (i) and (ii)
with calculus
F=dp/dt=d(mv)/dt= m. dv/dt= ma
with no external force a=o and linear momentum is conseved
I get how this works similiarly with rotational dynamics only in COLLISION
in the first method, conservation of angular momentum closed system durning collsion can be dudeuced by the 1st method by replacing
W_1 by torque by exteral force
F_12 by T_12= torque created on body(or disc) 1 by body 2
m.v_f1 by I.w_f1= angular momentum of body after collision
m.v_01 by I.w_01= angular momentum of body before collision
and by calculus
T= dL/dt = d(Iw)/dt =I. dw/dt = Ia
this is what i don't get: in the case of a ice skater, pulling arms in during rotation, I(moment of inertia) is changed, so how is conservation of momentum explained with calculus method when I of the system changes and i don't get at all how the non calculus method explains the conservation in this case (i only get it if it is during impact in rotational but in linear if totally understand how this method explains the conservation whether it impact of no impact)
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